On the Simplex method for 0/1 polytopes

TL;DR

This paper introduces new pivot rules for the Simplex method tailored for 0/1 polytopes, achieving strongly polynomial or linear bounds on steps, with optimal bounds on key combinatorial polytopes, based on geometric analysis.

Contribution

It proposes novel pivot rules for the Simplex method that guarantee strongly polynomial or linear step bounds on 0/1 polytopes, improving efficiency.

Findings

Number of steps is strongly polynomial or linear in dimension or variables.

Bounds are asymptotically optimal on key combinatorial polytopes.

Connections established between pivot rules and other algorithms.

Abstract

We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the dimension or in the number of variables. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1 polytopes and novel modifications to the classical Steepest-Edge and Shadow-Vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization.